Optimal. Leaf size=122 \[ -\frac{1}{20} \left (\frac{5 d (d+e x)^4}{e^2}-\frac{4 (d+e x)^5}{e^2}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{b d^5 n \log (x)}{20 e^2}+\frac{1}{15} b d^2 e n x^3+\frac{b d^4 n x}{5 e}+\frac{3}{20} b d^3 n x^2+\frac{1}{80} b d e^2 n x^4-\frac{b n (d+e x)^5}{25 e^2} \]
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Rubi [A] time = 0.0911198, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {43, 2334, 12, 80} \[ -\frac{1}{20} \left (\frac{5 d (d+e x)^4}{e^2}-\frac{4 (d+e x)^5}{e^2}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{b d^5 n \log (x)}{20 e^2}+\frac{1}{15} b d^2 e n x^3+\frac{b d^4 n x}{5 e}+\frac{3}{20} b d^3 n x^2+\frac{1}{80} b d e^2 n x^4-\frac{b n (d+e x)^5}{25 e^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2334
Rule 12
Rule 80
Rubi steps
\begin{align*} \int x (d+e x)^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac{1}{20} \left (\frac{5 d (d+e x)^4}{e^2}-\frac{4 (d+e x)^5}{e^2}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{(d+e x)^4 (-d+4 e x)}{20 e^2 x} \, dx\\ &=-\frac{1}{20} \left (\frac{5 d (d+e x)^4}{e^2}-\frac{4 (d+e x)^5}{e^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{(b n) \int \frac{(d+e x)^4 (-d+4 e x)}{x} \, dx}{20 e^2}\\ &=-\frac{b n (d+e x)^5}{25 e^2}-\frac{1}{20} \left (\frac{5 d (d+e x)^4}{e^2}-\frac{4 (d+e x)^5}{e^2}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{(b d n) \int \frac{(d+e x)^4}{x} \, dx}{20 e^2}\\ &=-\frac{b n (d+e x)^5}{25 e^2}-\frac{1}{20} \left (\frac{5 d (d+e x)^4}{e^2}-\frac{4 (d+e x)^5}{e^2}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{(b d n) \int \left (4 d^3 e+\frac{d^4}{x}+6 d^2 e^2 x+4 d e^3 x^2+e^4 x^3\right ) \, dx}{20 e^2}\\ &=\frac{b d^4 n x}{5 e}+\frac{3}{20} b d^3 n x^2+\frac{1}{15} b d^2 e n x^3+\frac{1}{80} b d e^2 n x^4-\frac{b n (d+e x)^5}{25 e^2}+\frac{b d^5 n \log (x)}{20 e^2}-\frac{1}{20} \left (\frac{5 d (d+e x)^4}{e^2}-\frac{4 (d+e x)^5}{e^2}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.118846, size = 130, normalized size = 1.07 \[ d^2 e x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{2} d^3 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{3}{4} d e^2 x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{5} e^3 x^5 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{3} b d^2 e n x^3-\frac{1}{4} b d^3 n x^2-\frac{3}{16} b d e^2 n x^4-\frac{1}{25} b e^3 n x^5 \]
Antiderivative was successfully verified.
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Maple [C] time = 0.225, size = 598, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.984495, size = 190, normalized size = 1.56 \begin{align*} -\frac{1}{25} \, b e^{3} n x^{5} + \frac{1}{5} \, b e^{3} x^{5} \log \left (c x^{n}\right ) - \frac{3}{16} \, b d e^{2} n x^{4} + \frac{1}{5} \, a e^{3} x^{5} + \frac{3}{4} \, b d e^{2} x^{4} \log \left (c x^{n}\right ) - \frac{1}{3} \, b d^{2} e n x^{3} + \frac{3}{4} \, a d e^{2} x^{4} + b d^{2} e x^{3} \log \left (c x^{n}\right ) - \frac{1}{4} \, b d^{3} n x^{2} + a d^{2} e x^{3} + \frac{1}{2} \, b d^{3} x^{2} \log \left (c x^{n}\right ) + \frac{1}{2} \, a d^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.02135, size = 394, normalized size = 3.23 \begin{align*} -\frac{1}{25} \,{\left (b e^{3} n - 5 \, a e^{3}\right )} x^{5} - \frac{3}{16} \,{\left (b d e^{2} n - 4 \, a d e^{2}\right )} x^{4} - \frac{1}{3} \,{\left (b d^{2} e n - 3 \, a d^{2} e\right )} x^{3} - \frac{1}{4} \,{\left (b d^{3} n - 2 \, a d^{3}\right )} x^{2} + \frac{1}{20} \,{\left (4 \, b e^{3} x^{5} + 15 \, b d e^{2} x^{4} + 20 \, b d^{2} e x^{3} + 10 \, b d^{3} x^{2}\right )} \log \left (c\right ) + \frac{1}{20} \,{\left (4 \, b e^{3} n x^{5} + 15 \, b d e^{2} n x^{4} + 20 \, b d^{2} e n x^{3} + 10 \, b d^{3} n x^{2}\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.67076, size = 218, normalized size = 1.79 \begin{align*} \frac{a d^{3} x^{2}}{2} + a d^{2} e x^{3} + \frac{3 a d e^{2} x^{4}}{4} + \frac{a e^{3} x^{5}}{5} + \frac{b d^{3} n x^{2} \log{\left (x \right )}}{2} - \frac{b d^{3} n x^{2}}{4} + \frac{b d^{3} x^{2} \log{\left (c \right )}}{2} + b d^{2} e n x^{3} \log{\left (x \right )} - \frac{b d^{2} e n x^{3}}{3} + b d^{2} e x^{3} \log{\left (c \right )} + \frac{3 b d e^{2} n x^{4} \log{\left (x \right )}}{4} - \frac{3 b d e^{2} n x^{4}}{16} + \frac{3 b d e^{2} x^{4} \log{\left (c \right )}}{4} + \frac{b e^{3} n x^{5} \log{\left (x \right )}}{5} - \frac{b e^{3} n x^{5}}{25} + \frac{b e^{3} x^{5} \log{\left (c \right )}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20664, size = 230, normalized size = 1.89 \begin{align*} \frac{1}{5} \, b n x^{5} e^{3} \log \left (x\right ) + \frac{3}{4} \, b d n x^{4} e^{2} \log \left (x\right ) + b d^{2} n x^{3} e \log \left (x\right ) - \frac{1}{25} \, b n x^{5} e^{3} - \frac{3}{16} \, b d n x^{4} e^{2} - \frac{1}{3} \, b d^{2} n x^{3} e + \frac{1}{5} \, b x^{5} e^{3} \log \left (c\right ) + \frac{3}{4} \, b d x^{4} e^{2} \log \left (c\right ) + b d^{2} x^{3} e \log \left (c\right ) + \frac{1}{2} \, b d^{3} n x^{2} \log \left (x\right ) - \frac{1}{4} \, b d^{3} n x^{2} + \frac{1}{5} \, a x^{5} e^{3} + \frac{3}{4} \, a d x^{4} e^{2} + a d^{2} x^{3} e + \frac{1}{2} \, b d^{3} x^{2} \log \left (c\right ) + \frac{1}{2} \, a d^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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